Which will be = o, becaufe the firft and laft terms are the 

 fame with contrary ligns, and becaufe o will be a fador in each 

 of the other terms. That the firft and laft terms will have con-" 

 trary figns appears from confidering that in the laft term there 

 are n — i negative factors, and confequently when n is even the 

 produd will be negative and the fign of the term itfelf will be 

 pofitive becaufe m + i (n-\- i) is odd, and when n is odd the 

 produdt will be pofitive and the lign of the term negative. 



3. Substituting for w, 2, the general term of the firft hori- 

 zontal rank = + «- 



n — I T 



n — 2. 2 |- = o. 



«— 3 -3 



+ «— 3 



From thefe different conclufions wc colled: ift, that (be- 

 caufe ^L = m — I. H + K + K') if each of the terms in any hori- 

 zontal rank = o the terms in the rank below are equal : zdly, 

 therefore it follows becaufe a term in each rank = o (when 

 m = n) that if each of the terms in any horizontal rank are equal 

 to o, that the terms of the rank beneath are each = o, and 3dly, 

 tecaufe thofe of the firft horizontal rank are each = o, it follows 

 therefore that each term of the table = o. Q; E. D. 



Theorem. 



I. Let the circumference of a circle be divided into n equal 

 parts 00', O'O", &c. and from a point P in the radius OC or 



U 2 the 



